Arithmetic aspect of operator algebras

C ^OMPOSITIO M ^ATHEMATICA

R. J. P LYMEN

C. W. L EUNG

Arithmetic aspect of operator algebras

Compositio Mathematica, tome 77, n^o3 (1991), p. 293-311

<http://www.numdam.org/item?id=CM_1991__77_3_293_0>

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Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques

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Arithmetic

aspect

^of

operator algebras

R.J. PLYMEN and C.W. LEUNG

©

Department of Mathematics, The University, Manchester M13 9PL, U.K.

Received 8 August 1989; accepted in revised form 1 May ¹⁹⁹⁰

1. Introduction

1.1. The classical notion of a cusp form f in the upper half-plane leads first to the concept ^{of a} cusp form ^on the adele _group of GL(2) ^over Q, ^{and thence to the}

idea of an automorphic cuspidal representation 03C0f ^{of the} adele _{group of} GL(2).

We recall that the adele group of GL(2) ^{is the} restricted product ^{of the local}

groups GL(2,

Qp)

^{where p is a place of Q. If p is infinite then}

Qp

^{is the real field R;}

if _p is finite then

Qp

^{is the} p-adic ^{field. The} unitary

representation nf

^{may be} expressed ^as ~03C0p ^{with one local} representation 1tp for each local group

GL(2, Qp).

^{It is in} this way that the unitary representation theory ^of p-adic

groups such ^as GL(2,

Qp)

^{enters into the modern} theory ^of automorphic ^forms [6,18].

A classical problem ⁱⁿ C*-algebra theory ^{is to determine the structure of C*-} algebras constructed from groups. In this article we show how unitary representation theory ^{can be used to} elucidate the structure of the reduced C*-

algebra C*(G) ^{when G is a} p-adic Chevalley group.

The algebra C*(G) ^{is defined as follows. We choose a} left-invariant Haar

measure on G, ^{and form} the Hilbert _space

L2(G).

^{The left} regular representation

03BB of

L’(G)

^on

L2(G)

^is given by (Â(f»(h) ⁼ .Î ^{* h}

where

f ~ L1(G), h~L2(G)

^{and * denotes} convolution. The C*-algebra generated by ^the image of 03BB is the reduced C*-algebra C*(G).

1.2. Before describing ^our results, ^{we turn back to the case} of linear connected reductive Lie groups. The examples ^{to bear in mind are the} identity component of the general linear group GL(n, R) ^{and the} special linear group SL(n, R).

Wassermann [17] has determined the structure of C*(G), up ^to stable

isomorphism, ^{as a direct sum of} component C*-algebras. ^Each component ^{is the} crossed product ^{of an abelian} C*-algebra by ^{a finite} group. The abelian C*-

algebra ^is

Co(Â/W’(r»)

^{with  the} unitary ^{dual of the} split component A ^{of a Levi}

factor M. The finite group is the Knapp-Stein R-group R(i) ^{of a} representation ⁱ

in the discrete series of the °M subgroup ^{of a Levi factor M. The} groups W’(1)

and R(T) ^{are related} by W(i) = W’(i). R(i) (semidirect product) ^where W(03C4) ^{is the} isotropy subgroup ^{of the} appropriate Weyl group W(M) = NG(M)/M. ^{The R-}

group R(i) controls the reducibility ^{of the induced} representation iGM(i),

obtained by ^first extending i across a parabolic subgroup ^{P with} Levi factor M and then unitarily inducing ^{from P to G. In fact the} commuting algebra ^of iGM(T)

is isomorphic ^{to the group} algebra of the finite _group R(T). ^Each component ^C*- algebra ^{is of the form}

C0(Â/W’(03C4))

R(i) ^{with r in} the discrete series of 0 M.

Reducibility ^{of the} tempered representation iGM (i) ^will prevent ^the component C*-algebra ^from being ^abelian. Irreducibility ^of iGM(T) will result in an abelian

C*-algebra

Co(Â/W(T».

The classic example ^{of this} phenomenon ^is provided by

G ⁼ SL(2, R). ^{Here the °M} subgroup ^{with M} minimal is the 2-element _group Z/2.

The trivial representation ^of Z/2 gives ^{rise to the} C*-algebra Co(R) ^which corresponds ^{to the even} principal series of SL(2, R); the non-trivial _represen- tation of Z/2 gives ^{rise to the crossed} product Co(R) Z/2 ^which corresponds ^to

the odd principal series. The crossed product ^{arises because the induced} representation iGM(03C4) ^{has 2} irreducible components ^{when T is} the non-trivial

representation ^of Z/2. This is the unique ^reducible representation ^{in the} unitary principal ^{series of} SL(2, R).

1.3. We now turn to the case of reductive p-adic groups. The theory ^{of the R-}

group and the normalization of standard intertwining operators ^{has now} reached a rather advanced stage ^of development [19]. ^There is, ^{in the} background, ^the optimism ^{of the Lefschetz} principle ^as formulated by ^Harish- Chandra, ^which says that whatever is ^true for real reductive groups is also ^true for p-adic groups. One might hope ^{to obtain a structure theorem for} the reduced

C*-algebra ^{of reductive} p-adic groups modelled ^on Wassermann’s result.

In order to focus on the difficulty, ^we shall consider p-adic Chevalley groups.

We recall that a p-adic Chevalley group is semisimple. ^We shall also restrict ourselves to the case of minimal Levi sugroups M. In this case a minimal Levi

subgroup ^{is a maximal torus T in} G, ^{and we have the basic} decomposition

in the notation of Steinberg [16]. ^The group A is a finitely generated free abelian group whose rank is the parabolic ^{rank of G. Its} unitary ^dual A is a compact

torus of dimension equal ^{to this rank. The} Weyl group of G, namely NG(T)/T,

acts on the compact ^torus Â. Here one can pinpoint the drastic difference from the case of real Lie groups. In the case of real Lie _groups, the Weyl group ^acts

linearly ^{on the real vector} space Â. In the p-adic case, A is a compact ^{torus. The} Weyl group ^{acts as} automorphisms

of Â,

^{but it cannot of course act} linearly. ^{It is}

true that each isotropy subgroup ^acts linearly ^{on the} tangent space ^at each point

of

A,

but this does not seem to help. ^The non-linearity of the action of the Weyl

group ^means that we cannot adapt ^the proof ^of Wassermann to the p-adic ^case.

To compensate ^for non-linearity, ^{we have to} supply ^a sufficient condition on the

representation ^{T. The} representation T ^{is in the} discrete series of ° T; ^{but since ° T}

is compact, ^{this means} any unitary ^{character of or.}

1.4. We now describe this condition. We consider the root system underlying

the p-adic Chevalley group G. Let ^a be a root and a v be a co-root. Definer,,, by

the equation Ta(U) ⁼ 03C4(03B1v (u))

for all p-adic ^{units u. Our} condition is that

(*) Ta =1= ^{1 for all a} &#x3E; 0 and W(i) ^{is abelian.}

In this case we can recover a component C*-algebra ^{of the form}

exactly ^as in the real case. The condition (*) ^{is the exact} analogue ^{of an essential} representation ^{in the discrete series of} °M, ^{as in} [17].

We illustrate this condition in the case of the special linear group SL(n). ^The

standard maximal torus of SL(n) comprises ^all elements of the form

diag(x1, x2,..., xn) ^with x1x2 ··· Xn ⁼ 1. The subgroup ^{° T} comprises all elements of the form diag(u1, U2, ... , un) ^with u1u2 ··· Un ⁼ 1 and the uj ^{are all} p-adic ^units.

A unitary character i of ° T is necessarily ^{of the form}

2: diag(ut, ^..., un) ^t-+ ~1(u1) ··· ~n(un)

with

each ~j

^{a unitary character of the group of p-adic} units. The unitary

character i has projective coordinates (~1 :···: Xn). ^{The condition} (*) ^{is then that}

all coordinates ~1,···, ~n ^are distinct.

The unitary ^{dual of the} group U of p-adic units may be identified with the group of all pmth ^{roots of 1 for m} = 1, 2, 3, ^{.... So each character} _Xj ^has prime

power order.

The situation is sufficiently ^well illustrated by ^the p-adic group SL(10). Let X

be a unitary ^{character of F} of order 5. Such characters will exist if and only ^if

p ⁼ 5 _{or p -} 1 mod 5. The arithmetic of the field enters at this point. ^{Let the} unitary character T have co-ordinates

where y is another character independent ^of x. The Weyl group of SL(10) ^{is the} symmetric group Slo ^{and the} isotropy subgroup W(i) ^{is the} cyclic group Z/5.

But condition (*) ^is satisfied and so the character r contributes a component ^C*- algebra

where A is a 4-dimensional compact ^{torus. This torus} may be identified with

T5/T

^with the group Z/5 acting by cyclic permutation. ^{In this case} the induced

representation iGM(03C4) ^{is a} unitary representation ^{with 5} distinct irreducible components. ^Once again, ^the reducibility prevents ^the component C*-algebra

from being ^abelian.

The condition (*) ^is conspicuously ^{not met} by ^the unramified unitary principal series of the p-adic group SL(2). ^For here the character i has coordinates (1 : 1).

1.5. One might hope ^{for a} rigid relationship ^{between the} arithmetic of the local field F and the structure of the component C*-algebras of the reduced C*-

algebra. ^{There is one central} example ⁱⁿ which this is definitely ^{true. For}

consider the p-adic Chevalley group SL(l) ^{with 1} prime ^and 1 dividing p - 1. In this case there are 1 ^- 1 component C*-algebras stably isomorphic ^{to the}

crossed product

where T is the unit circle and 7L/l ^acts by cyclic permutation. ^{The 1 fixed} points

have an arithmetic significance. ^{Each fixed} point ^Q is associated to a cyclic ^of

order 1 totally ^{ramified extension} field, and all such extension fields are

accounted for in this way. We recall that ^an extension field E of F is totally ramified if the valuation on E satisfies

where val denotes valuation. Since iGM(03C3) ^has 1 distinct irreducible components, there is at this point ^a rigid relationship ^{between the} arithmetic of the field, ^the

structure of the crossed product

C(Tl/T)

Z/l, ^{and the} reducibility ^{of the} tempered representation iGM(U).

1.6. In Section 2 of this article, ^{we unravel the} proof in Wassermann [17, p. 560]

and then adapt ^{it to the} p-adic ^{case. The main} difference, ^{as we have} already explained, ^{is to} replace ^{the real vector} space A _by a compact ^torus S, ^to replace

the linear action of the appropriate Weyl group ^on A by ^a non-linear action on

S, ^{and then to} compensate ^{for this} non-linearity. ^{We must} emphasize ^{that this}

section is modelled closely ^on Wassermann’s proof.

To convert the standard intertwining operators ^{into a} unitary 1-cocycle

involves solving ^a difficult normalization problem. Shahidi has recently proved

a conjecture of Langlands ^on normalization of intertwining operators by ^means

of local Langlands ^root numbers and L-functions, ^at least when the _{group is}

quasi-split ^{and the} inducing representation ^is generic [19, ^Theorem 7.9]. ^{In the} special ^case of the minimal unitary principal ^{series of a} p-adic Chevalley group

G, the normalization was dealt with by Keys [7]. ^{Fix a Borel} subgroup ^{B of G.}

Keys’ method involves Fourier analysis ^{on the} unipotent ^radical of the Borel

subgroup opposed ^to B, and ^a suitable version of the gamma function. This

converts the standard intertwining operators ^{into a} unitary 1-cocycle. This,

taken in conjunction ^{with condition} (*), ^{allows us to} apply ^the C*-algebra

results of Section 2. The resulting C*-algebras ^{are crossed} products of the form

C(S) R(r).

In Section 4, ^we compute ^all R-groups ^{which can arise in this context. This}

section depends ^on Key’s classification of R-groups [7].

In Section 5, ^we illustrate the arithmetic aspect ^{in the case of the} special linear

group SL(4. ^{This section} hinges ^{on the Artin} reciprocity ^{law in} local class field

theory.

2. On certain fixed-point algebras

2.1. Let A be a C*-algebra, ^{let r be a} finite group, and let (A, r, a) ^{be a C*-} dynamical system. ^Let C(r, A) ^be the linear _space of all _maps from r to A. We shall denote the crossed product ^{of A} by ^{r as A} 03B1 r.

Define p by p(t) ^{= I for all t} in r. Then _p is a projection ^{in the} multiplier algebra M(A 03B1 r). ^{Let Aa be the} fixed-point algebra ^{of A. We embed A03B1 in}

A _03B1 r by sending ^{x to the constant} function whose value is x. The image ^{of this} embedding ^is precisely p(A 03B1 r)p ^{so we have}

2.2. Suppose that t ~ ut is a map from r into the unitary group of M(A) ^such

that

and let 03B2t=(Ad ut)03B1t. The map t H ut ^{is a} unitary 1-cocycle ^{and the C*-} dynamical systems (A, r, a) ^and (A, r, 03B2) ^{are exterior} equivalent [10, p. 357]. ^The map 03A6 ^defined by 03A6(y)(t) ⁼ y(t)u*t secures an isomorphism of crossed products

Let q ^{be the} map from r into M(A) ^defined by q(t) ⁼ u,. Then

4)(q)(t) ⁼ q(t)u* ⁼ u,u* ^{= I. We} therefore have

The fixed-point algebra ^with respect to Pis isomorphic ^{to a corner} of the crossed

product ^with respect ^{to a.}

2.3. An element b in a C*-algebra A ^is strictly positive ^if 9(b) &#x3E; 0 for every ^state 9 of A. Let e be a projection ^{in the} multiplier algebra of A and let  be the

unitary dual of A. The support supp(e) ^{is defined as the set of all n in} Â such that

03C0(e) ~ ^{0. A corner of A is a} hereditary subalgebra ^{of A of the form} eAe where e is

a projection ⁱⁿ M(A). ^{A corner is} full ^{if it is not contained} in any proper closed 2- sided ideal of A. Two C*-algebras ^{A and B are} stably isomorphic ^if

A (D k ^B Q k where k is the standard C*-algebra ^of compact operators. ^These ideas are intimately related and we shall need the following ^result.

2.4. LEMMA. Let A be a C*-algebra ^{with a} strictly positive ^{element. Let e} and f

be projections ⁱⁿ M(A) ^{with the same} support. Then ^{eAe and} f Af ^are stably isomorphic.

Proof. ^{Let I be} the closed 2-sided ideal of A generated by ^{eAe and} f Af. ^One

first checks by methods of Dixmier [5, ^Section 3.2] ^{that eAe is a full corner of 7.}

Now I is a C*-algebra ^{with a} strictly positive element and each of eAe and f Af

is a full corner of 7. Therefore eAe and f Af ^are stably isomorphic, by ^{results in} [1, Corollary 2.9) ^and [2].

2.5. From now on, S will denote a compact ^{torus. We shall} regard S ^{as a} compact ^{abelian Lie} group. We shall exploit the fact that Pontryagin duality

works just ^as well for S as it does for a real finite-dimensional vector space, ^as in

[17, p. 560].

2.6. We shall suppose that the finite group r ^{acts as} automorphisms ^{of S. The} unitary ^dual S is ^a finitely generated free abelian group. The unitary ^dual of S is canonically isomorphic ^{to S} by Pontryagin duality. ^The action of r on S determines an action of r on

S,

^{and an action a of 0393 on} C(S). ^{We have the basic}

Fourier transform

where 0393 denotes semidirect product.

Since each r-orbit in S is finite, ^{and S is a} complete separable ^metric space, ^we

can use the Mackey machine. This gives ^{us a} prescription ^{for the} unitary ^{dual of}

the countable discrete group r. Let y ^{E S and let}

ry

^be the stabilizer of _y.

Form the semidirect product

0393y.

^{Let 6 be an} irreducible representation ^of ry. ^{Then y 0 (7 is an} irreducible representation ^of

T’y.

Induce from

ry

^to

r ^to obtain the irreducible representation 03C0y,03C3. ^{We shall} regard TCy,a ^{as an} irreducible representation of the crossed product C(S)03B1^r.

Fix ₉ in C(S) and to ^{in r. Define F in} C(S)03B1 ^r by setting F(to) = ^{qJ and} F(t) ^{= 0 if t ~} to. ^The character of 6 will be denoted _la.

2.7. LEMMA. The trace of

TCy,a(F)

^{is given by}

The summation is over all r in R and all r-1 sr in

r y

^{where R is a set of coset}

representatives of

r /r y.

Proof. ^The proof ^{is a tedious} computation using ^the Fourier transform

identification C(S) 03B1 0393 ~

C*(

r) ^and the standard formula for the induced character ^as in Serre [14].

2.8. Let Mn(C) ^{be the} algebra ^{of n x n} complex ^{matrices. The action a of r on} C(S) ^{is extended to an action a on} C(S) ~ Mn(C) ^{as follows:}

(Xt(CP p T) = (,x,«p» Q ^{T. We} then have

The representation TCy,u ^extends uniquely ^{to an} irreducible representation ^of (C(S)03B10393) ~ Mn(e) ^{which we continue to denote} by 7ry’«. ^{If now} 9 ^E C(S, M,,(C», to E r and F is defined as F(to) = 9, F(t) ⁼ 0(t ~ to) ^{then we have} similarly

Let _u, E C(S, U(n)) ^{be a} unitary 1-cocycle, with t E r. Let q(u) ^{be the} map from r into C(S) p Mn(C) ^defined by q(u)(t) ⁼ u, for all t in r. Then q(u) ^{is a} projection ⁱⁿ (C(S) ⁰ Mn(C))03B1 0393. ^{A routine} computation ^{based on 2.7 will now} yield ^the trace-multiplicity ^formula

Trace(03C0y,03C3)q(u) = 03C8y,

^u)

where

03C8y(t)

⁼ u,(y) ^{for all t in the} isotropy subgroup

ry

and a is the conjugate ^of

the representation ^{J. That is to} say, the rank of the projection

03C0y,03C3(q(u))

^is equal

to the multiplicity ^{with which à occurs in}

t/1 Y"

2.9. From now on, ^we shall take our C*-algebra ^{A to be} C(S) ^Q k(H). ^Our unitary 1-cocycle ^{t ~} ut will be a map from r into the unitary group C(S, U(H)).

Define

Then t _{~ vt} is a unitary 1-cocycle. ^In fact, ^t ~ vt is a unitary representation ^{of 0393.}

Define, ^{as in} (2.2)

We shall suppose that there exists an increasing sequence (en) ^of finite-rank

projections ⁱⁿ L(H) which converges strongly ^{to I and commutes with each} u,.

Let

03C8ny(t)

^{be the} compression ^of ut(y):

for each natural number n, each y ⁱⁿ S, ^{each t in the} isotropy subgroup

ry.

Suppose further that for each y ^{in S} there exists N such that

03C8ny

^{and 03C8n0|}

ry

^are quasi-equivalent ^{whenever n} &#x3E; N. This means that

03C8ny

^and

03C8n0 ^ry,

^{as unitary}

representations ^{of the} isotropy subgroup

ry,

^{have the same} irreducible _compo-

nents (though ^not necessarily ^{with the same} multiplicity). ^{So our} condition is

(*) 03C8n0|

0393y

^is quasi-equivalent ^to

03C8ny(n

&#x3E; N).

2.10. LEMMA. If (*) holds then the projections q(u) ^and q(v) ^{have the same} support.

Proof. ^Let qn(u) ^and qn(v) ^{be the} projections ⁱⁿ M((C(V) ^Q k)03B1 r) ^defined by

for all t in r. Note that qn(u)) ^{is an} increasing sequence of projections. ^Since

en ^- I strongly, it follows that n(q,,(u» ^~ n(q(u)) strongly ^for any representation

of (C( ) ⁰ k) ^{r. Therefore}

Therefore supp(q(u)) ⁼ supp(q(v)).

2.11. LEMMA. Let A = C(S) Q ^{k and let a be the action} of ^r given by a, «p ⁰ T) = (Xt(qJ) ^{0 T. Let t ~} u, be a unitary 1-cocycle ^with u, in C(S, U(H)) ^and

let

If (*) holds then AO is stably equivalent ^{to A Y.}

Proof. Note first that (A, r, 03B1), (A, r, 03B2) ^and (A, r, y) ^{are exterior} equivalent C*-dynamical systems. By (2.2) ^{we have}

and

Since C(S) ^{is a unital} C*-algebra, ^{A and A} 03B1 r each contains a strictly positive

element. Therefore, by (2.4) ^and (2.10), ^{Ali and At’ are} stably isomorphic.

2.12. Now t _{~ vt} is a unitary representation ^{0393 ~} U(H). ^{Let r’} comprise ^{all t in r}

such that v, ^is scalar, ^so that r’ is a normal subgroup of 0393. We shall suppose that

(**) ^{r admits a} complementary subgroup r 1 such that r is a semidirect

product 0393’ 03931.

(***) ^The unitary representation ^of ri given by t ~ vt is quasi-equivalent ^{to the}

left regular representation ^{À of} 03931 ^on e(r 1).

2.13. THEOREM. Let (A, r, 03B2) ^{be the} C*-dynamical system ^under discussion. If

conditions (*), (**) ^and (***) hold then the fixed point algebra ^{AO is} stably isomorphic ^{to the crossed} product C(S/r’) r 1.

Proof. ^{We have}

whenever t lies in the normal subgroup ^{r’. Now}

where subscripts 03931 (resp. r’) signify restrictions of y ^to 03931 (resp. r’). ^{We have}

where ôi, is the action induced on the quotient space S/r’ ^and r ~ 03931. By ^Takai duality ^{we have}

where, ^{for all r in} 03931,

By (* * *) ^{it is now} clear that Ay is stably isomorphic ^to C(S/r’) ^{r 1.}

The theorem now follows from Lemma (2.11).

3. Minimal unitary principal ^{series for} p-adic Chevalley groups 3.1. Throughout ^this section,

Q p

^{denotes a p-adic field. Let}

Now

where (91

= {1

^{+ x:}

Ixl, 1}

^and 03B5&#x3E; ~ Z/(p - 1) ^{as in} [13].

Let L be ^a complex semi-simple ^Lie algebra ^{and A be its root} system.

Throughout ^this section, ^{G denotes a} p-adic Chevalley group determined by ^L.

The notations x03B1(t), h03B1(t), w03B1(t) ^{are defined as in} [16]. ^Let X03B1 ⁼

{x03B1(t): t ~ Qp}.

Then G is generated by ^all X a, ^{a E A. Let T be the} subgroup ^{of G} generated by ^all ha(t). ^{Then T is a maximal torus of G. Denote the} Weyl group N G(T)/T by ^{W. Let}

U be the subgroup ^{of G} generated by ^all X03B1, ^{where oc is a} positive ^{root. Then we}

have the following ^Levi decomposition [16]:

Let B be the _group generated by ^{T and} U, ^{then B = T. U} (semi-direct product)

and U is normal in B. Moreover B is a Borel subgroup ^{of G. Let K be the} group

generated by

{x03B1(t):

03B1 ~ 03B4,

t ~ O}.

^{Then K is a} good ^maximal compact subgroup

of G. We have the Iwasawa decomposition G = KB = K T U (non-uniquely). ^Let

° T be the group generated

by {h03B1(t): 03B1 ~ 0394, t ~ O }

and let A be the _group

generated by

{h03B1(pn): 03B1 ~ 0394, n ~ Z}.

^{Then we have the direct} product decomposition

In fact, ^the decomposition T ^{= ° T. A is not canonical} (in contrast to real groups)

and depends ^{on the choice of} uniformizer that is here taken to be _p. For an

arbitrary p-adic field, ^{there is no} such canonical uniformizer. In fact, ^we only

have

with no canonical splitting.

Let Bf(T) ⁼

(ç

^E Hom(T, U(1)):

qJ(OT)

⁼

1}.

^{Note that 03A8(T) ~ Â. Let} X(T) ⁼ Hom(T,

Q p).

^Then X(T) ^{is a} free abelian group of finite rank.

3.2. LEMMA. The _map which sends (03BE, Â) ⁱⁿ X(T) ^{x R to the character}

of T ^{induces a} homomorphism of X(T) ~ R ^onto 03A8(T), ^{whose kernel is} X(7) ^{0 Z.}

Proof Steinberg [16] and Rodier [12]. ^{From the above} lemma, ^{we see that} 03A8(T) ^{is a} compact ^torus.

Now ^we define an action of the Weyl group W on T

If XE 1;

w ~ W, w.~ is

defined by

where _{xw is any} representative ^{of w in} NG(T).

3.3. The groups B, ^{U are defined as in} (3.1). ^Let (03C3, V) ^{be an} irreducible unitary representation ^{of B. Let} H(03C3) ^{be the} space of all smooth functions f : ^{G ~ C such}

that

where ô. ^{is the modular function of B. The factor}

03B4-1/2B

^{is used so that} unitary representations ^{induce to} unitary representations. Define the induced _represen- tation

IndB6

^{to be left} translation in H(6). ^{Note that}

IndGB03C3

^{is an admissible} unitary representation ^{of G. Let T be a} unitary ^{character of ° T and a} unitary

character of A. Define

Then _To, 03BB0 ^are unitary characters of B. By ^{abuse of} notation, ^{we also denote} io and 03BB0 by T ^and respectively. ^{Note that}

Ind%i

^and

IndGB(03BB03C4)

^{can be realized on}

the same Hilbert space for any À ^E 03A8(T), ^see Silberger’s ^book [15].

3.4. In this section, the results were proved by Keys [7]. Suppose ^{6 is a} unitary

character of T. Define the standard intertwining operators

where the integration ^{is over}

U n wllw-1,

weNG(1) ^{is a coset} representative ^for

w E W and f E H(6). ^{We normalize} A(w,u) by ^the gamma function 0393w(03C3). To

define 0393w(03C3), proceed ^{as follows. Fix a} non-trivial additive character _x of F. A gamma function 0393(03BB) ^is associated to each non-trivial quasi-character 03BB ^{of F .}

Define 0393(03BB) ^as

P.V. f ~(x)03BB(x)|x|-1

^{dx. In case 03BB is} unramified, this formula must be understood in terms of analytic continuation, ^{for details see Taibleson} [20, p.48]. Suppose ^{now that a is a} simple root, and let ^w be the basic reflection wa. Then 0393w(03C3) is defined as where aa is the unitary character of F " given by Ua = uoha. ^{If w is a} product ^{of basic} reflections, ^then 0393w(03C3) is defined by ^an

iterative formula which follows the 1-cocycle ^{relation, as in} Keys [7, p. 360]. ^Let

By analytic continuation, a(w, a) is defined for all unitary characters a. With suitable choice of coset representative ^{w, the} cocycle ^relation

holds for all _{w 1, w2} in W.

3.5. Let T and À be defined as in (3.3). By ^the Peter-Weyl ^{theorem we have the} orthogonal ^{direct sum over all} p in K

Let %

^{be the} isotypic component ^{of class p of}

(Ind%i)

K. ^Then

Since

(Indlr,

H(,c» ^{is an} admissible representation, ^then [3] ^{we have}

dim V.

^o0

for all _p in K. Let W ⁼

{w ~

^{W : w.i =}

03C4}.

By [15, 5.2.1] ^{we have}